Superluminal solution of the Einstein equation of gravitation for weak and static fields

The author’s previous papers on superluminal time dilation [E. J. Betinis, Phys. Essays 11, 81 (1998)] and the superluminal Schrödinger equation [E. J. Betinis, Phys. Essays 11, 311 (1998)] are mentioned in order to inform the reader of the existence of the superluminal theories developed to date. The superluminal time dilation is utilized in the derivation of the Lorentz transformations in order to show as a first step that the superluminal theory for physics is valid and can maintain the invariance of the metric. Next the most general Einstein equation of gravitation is reduced to the special case of finding the weak and static fields of the gravitational potential for the universe and one receding galaxy. This case is further simplified by assuming that the receding galaxy is only moving radially from the center of the universe. It is found that by restricting the moving galaxy to having a velocity less than the speed of light c (the subluminal case), the turning point or “stopping point” of the receding galaxy would fall far short of the distance of the known observed outermost galaxy. By Hubble’s law, it is shown that the outermost galaxy would have a velocity greater than the speed of light, so the subluminal restriction also contradicts Hubble’s law as well. The superluminal formulation shows that the turning point if it exists can occur at the present assumed location of the outermost galaxy where its velocity is greater than the velocity of light. However, since observations presently show that the outermost galaxy is still receding at a velocity greater than the velocity of light c, the universe does not, at present, seem to be contracting or to be oscillatory. In addition, then the superluminal treatment for finding the gravitational potential as it allows a velocity greater than the velocity of light is also consistent with the Hubble’s law calculation of the velocity of the outermost receding galaxy. In the above formulation of the weak and static gravitational fields of the Einstein equation, spacetime is not really totally flat but is slightly curved. In addition, the group theory aspects of the Lorentz transformations for both the subluminal and superluminal cases are demonstrated.